Homomorphism

Not to be confused with homeomorphism. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The word homomorphism comes from the Greek language: homos meaning "same" and morphe meaning "shape". Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Note the similar root word "homoios," meaning "similar," which is found in another mathematical concept, namely homeomorphisms. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

1 Informal discussion
2 Formal definition
3 Types of homomorphisms
4 Kernel of a homomorphism
5 Homomorphisms and e-free homomorphisms in formal language theory
6 See also
7 References

Informal discussion

Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. In Mathematics, an operator is a function which operates on (or modifies another function The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function These functions are known as homomorphisms.

For example, consider the natural numbers with addition as the operation. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an A function which preserves addition should have this property: f(a + b) = f(a) + f(b). For example, f(x) = 3x is one such homomorphism, since f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b). Note that this homomorphism maps the natural numbers back into themselves.

Homomorphisms do not have to map between sets which have the same operations. For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication. A function which preserves operation should have this property: f(a + b) = f(a) * f(b), since addition is the operation in the first set and multiplication is the operation in the second. Given the laws of exponents, f(x) = e^x satisfies this condition : 2 + 3 = 5 translates into e² * e³ = e⁵.

A particularly important property of homomorphisms is that if an identity element is present, it is always preserved, that is, mapped to the identity. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that Note in the first example f(0) = 0, and 0 is the additive identity. In the second example, f(0) = 1, since 0 is the additive identity, and 1 is the multiplicative identity.

If we are considering multiple operations on a set, then all operations must be preserved for a function to be considered as a homomorphism. Even though the set may be the same, the same function might be a homomorphism, say, in group theory (sets with a single operation) but not in ring theory (sets with two related operations), because it fails to preserve the additional operation that ring theory considers. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those

Formal definition

A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure; i. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, e. properties like identity elements, inverse elements, and binary operations. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two

N.B. Some authors use the word homomorphism in a larger context than that of algebra. Nota bene is a Latin phrase meaning "Note Well" coming from notāre —to note Some take it to mean any kind of structure preserving map (such as continuous maps in topology), or even a more abstract kind of map—what we term a morphism—used in category theory. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets This article only treats the algebraic context. For more general usage see the morphism article. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

For example; if one considers two sets $X$ and $Y$ with a single binary operation defined on them (an algebraic structure known as a magma), a homomorphism is a map $\phi: X \rightarrow Y$ such that

$\phi(u \cdot v) = \phi(u) \circ \phi(v)$

where $\cdot$ is the operation on $X$ and $\circ$ is the operation on $Y$ . In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure.

Each type of algebraic structure has its own type of homomorphism. For specific definitions see:

group homomorphism
ring homomorphism
module homomorphism
linear operator (a homomorphism on vector spaces)
algebra homomorphism

The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A homomorphism between two algebras over a field K, A and B, is a map FA\rightarrow B such that for all k Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In this setting, a homomorphism $\phi: A \rightarrow B$ is a map between two algebraic structures of the same type such that

$\phi(f_A(x_1, \ldots, x_n)) = f_B(\phi(x_1), \ldots, \phi(x_n))\,$

for each n-ary operation $f$ and for all $x i$ in $A$ .

Types of homomorphisms

An isomorphism is a bijective homomorphism. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Two objects are said to be isomorphic if there is an isomorphism between them. Isomorphic objects are completely indistinguishable as far as the structure in question is concerned.

An epimorphism is a surjective homomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

A monomorphism (also sometimes called an extension) is an injective homomorphism. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.

A homomorphism from an object to itself is called an endomorphism. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself

An endomorphism which is also an isomorphism is called an automorphism. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

The above terms are used in an analogous fashion in category theory, however, the definitions in category theory are more subtle; see the article on morphism for more details. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

Note that in the larger context of structure preserving maps, it is generally insufficient to define an isomorphism as a bijective morphism. One must also require that the inverse is a morphism of the same type. In the algebraic setting (at least within the context of universal algebra) this extra condition is automatically satisfied. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models"

Relationships between different kinds of homomorphisms.
H = set of Homomorphisms, M = set of Monomorphisms,
P = set of ePimorphisms, S = set of iSomorphisms,
N = set of eNdomorphisms, A = set of Automorphisms.
Notice that: M ∩ P = S, S ∩ N = A, P ∩ N = A,
M ∩ N \ A contains only infinite homomorphisms, and
P ∩ N \ A is empty.

Kernel of a homomorphism

Main article: Kernel (algebra)

Any homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b iff f(a) = f(b). In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" ↔ The relation ~ is called the kernel of f. It is a congruence relation on X. See Congruence (geometry for the term as used in elementary geometry The quotient set X/~ can then be given an object-structure in a natural way, i. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X e. [x] * [y] = [x * y]. In that case the image of X in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural Note in some cases (e. g. groups or rings), a single equivalence class K suffices to specify the structure of the quotient; so we can write it X/K. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X (X/K is usually read as "X mod K". The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure ) Also in these cases, it is K, rather than ~, that is called the kernel of f (cf. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism normal subgroup, ideal). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

Homomorphisms and e-free homomorphisms in formal language theory

Homomorphisms are also used in the study of formal languages. A formal language is a set of words, ie finite strings of letters, or symbols. ^[1] Given alphabets $Σ 1$ and $Σ 2$ , a function h : $\Sigma_1^*$ → $\Sigma_2^*$ such that $h (u v) = h (u) h (v)$ for all u and v in $\Sigma_1^*$ is called a homomorphism on $\Sigma_1^*$ . ^[2] Let e denote the empty word. If h is a homomorphism on $\Sigma_1^*$ and $h(x) \ne e$ for all $x \ne e$ in $\Sigma_1^*$ , then h is called an e-free homomorphism.

References

^ Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, ISBN 0 7204 2506 9. Seymour Ginsburg (1928-2004 was a pioneer of Automata theory Formal language theory and Database theory in particular and Computer science
^ In homomorphisms on formal languages, the * operation is the Kleene star operation. In Mathematical logic and Computer science, the Kleene star (or Kleene closure) is a Unary operation, either on sets of The $\cdot$ and $\circ$ are both concatenation, commonly denoted by juxtaposition. For concatenation of general lists see Append. In Computer programming, string concatenation is the operation of joining two character

A monograph available free online:

Burris, Stanley N. , and H. P. Sankappanavar, H. P. , 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

Dictionary

-noun

(algebra) A structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces.
(biology) A similar appearance of two unrelated organisms or structures

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